An Analog of the Salem-Zygmund Theorem for Orthogonal Polynomials on the Unit Circle
In this note, I study the result and proof of the classical Salem-Zygmund Theorem. I apply the method to random orthogonal polynomials on the unit circle. The goal is to find the distribution of M, the sup norm of suitably defined random polynomial orthogonal on the unit circle. In my proof, I use Bernstein and Chebyshev inequalities to achieve this goal. I find that for fixed large κ, the probability of M > κ drops significantly as the degree n of the orthogonal polynomial grows.